Journal of Digital Convergence (디지털융복합연구)

The Society of Digital Policy and Management (한국디지털정책학회)
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Using Tabu Search for L(2,1)-coloring Problem of Graphs with Diameter 2

Tabu Search를 이용한 지름이 2인 그래프에 대한 L(2,1)-coloring 문제 해결

  • Kim, SoJeong (Department of Applied Mathematics, Kongju National University) ;
  • Kim, ChanSoo (Department of Applied Mathematics, Kongju National University) ;
  • Han, KeunHee (Department of Applied Mathematics, Kongju National University)
  • 김소정 (공주대학교 응용수학과) ;
  • 김찬수 (공주대학교 응용수학과) ;
  • 한근희 (공주대학교 응용수학과)
  • Received : 2021.11.22
  • Accepted : 2022.02.20
  • Published : 2022.02.28
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Abstract

For simple undirected graph G=(V,E), L(2,1)-coloring of G is a nonnegative real-valued function f : V → [0,1,…,k] such that whenever vertices x and y are adjacent in G then |f(x)-f(y)|≥ 2 and whenever the distance between x and y is 2, |f(x)-f(y)|≥ 1. For a given L(2,1)-coloring c of graph G, the c-span is λ(c) = max{|c(v)-c(v)||u,v∈V}. L(2,1)-coloring number λ(G) = min{λ(c)} where the minimum is taken over all L(2,1)-coloring c of graph G. In this paper, based on Harary's Theorem, we use Tabu Search to figure out the existence of Hamiltonian Path in a complementary graph and confirmed that if λ(G) is equal to n(=|V|).

단순 무방향 그래프 G 의 L(2,1)-coloring은 d(u,v)가 두 정점 사이의 거리일 때 두 가지 조건 (1) d(x,y) = 1 라면 |f(x)-f(y)|≥ 2, (2) d(x,y) = 2 라면 |f(x)-f(y)|≥ 1 을 만족하는 함수 f : V → [0,1,…,k]를 정의하는 것이다. 임의의 L(2,1)-coloring c 에 대하여 G 의 c-span 은 λ(c)=max{|c(u)-c(v)|| u,v∈V} 이며, L(2,1)-coloring number 인 λ(G)는 모든 가능한 c 에 대하여 λ(G) = min{λ(c)} 로 정의된다. 본 논문에서는 Harary의 정리에 기반하여 지름이 2인 그래프에 대하여 여그래프에 해밀턴 경로의 존재여부를 Tabu Search를 사용해 판단하고 이를 통해 λ(G)가 n(=|V|)과 같음을 분석한다.

Keywords

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